Fixed point index in symmetric products

Salazar, José

doi:10.1090/S0002-9947-04-03533-0

Fixed point index in symmetric products
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by José M. Salazar

Trans. Amer. Math. Soc. 357 (2005), 3493-3508

DOI: https://doi.org/10.1090/S0002-9947-04-03533-0

Published electronically: September 2, 2004

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Abstract:

Let $U$ be an open subset of a locally compact metric ANR $X$ and let $f:U \rightarrow X$ be a continuous map. In this paper we study the fixed point index of the map that $f$ induces in the $n$-symmetric product of $X$, $F_{n}(X)$. This index can detect the existence of periodic orbits of period $\leq n$ of $f$, and it can be used to obtain the Euler characteristic of the $n$-symmetric product of a manifold $X$, $\chi (F_{n}(X))$. We compute $\chi (F_{n}(X))$ for all orientable compact surfaces without boundary.

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